A general analysis of exact nonlinear lumping is presented. This analysis can be applied to the kinetics of any reaction system with n species described by a set of first-order ordinary differential equations dy/dt=f(y), where y is an n-dimensional vector and fly) is an arbitrary n-dimensional function vector. We consider lumping by means of an ($) over cap n(($) over cap n less than or equal to n)-dimensional arbitrary transformation ($) over cap y=h(y). The lumped differential equation system is d ($) over cap y/dt=h(y)(($) over bar h(($) over cap y))f(($) over bar h(($) over cap y)), where h(y)(y) is the Jacobian matrix of h(y), h is a generalized inverse transformation of h satisfying the relation h(($) over bar h)=I-$ over cap n. Three necessary and sufficient conditions of the existence of exact nonlinear lumping schemes have been determined. The geometric and algebraic interpretations of these conditions are discussed. It is found that a system is exactly lumpable by h only if h(y)=0 is its invariant manifold. A linear partial differential operator A=Sigma(i=1)(n) f(i)(y)partial derivative/partial derivative y(i), corresponding to dy/dt=f(y) is defined. Using the eigenfunctions and the generalized eigenfunctions of A, the operator can be transformed to Jordan or diagonal canonical forms which give the lumped differential equation systems without determination of ($) over bar h. These approaches are illustrated by a simple example. The results of this analysis serve as a theoretical basis for the development of approaches for approximate nonlinear lumping.